Lambda calculus L9 103

Transcription

1 Lambda calculus L9 103

2 Notions of computability L9 104 Church (1936): λ-calculus Turing (1936): Turing machines. Turing showed that the two very different approaches determine the same class of computable functions. Hence: Church-Turing Thesis. Every algorithm [in intuitive sense of Lect. 1] can be realized as a Turing machine.

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4 λ-terms, M L9 105 are built up from a given, countable collection of variables x, y, z,... by two operations for forming λ-terms: λ-abstraction: (λx.m) (where x is a variable and M is a λ-term) application: (M M ) (where M and M are λ-terms). Some random examples of λ-terms: x (λx.x) ((λy.(x y))x) (λy.((λy.(x y))x))

5 λ-terms, M L9 105 Notational conventions: (λx 1 x 2... x n.m) means (λx 1.(λx 2...(λx n.m)...)) (M 1 M 2... M n ) means (...(M 1 M 2 )... M n ) (i.e. application is left-associative) drop outermost parentheses and those enclosing the body of a λ-abstraction. E.g. write (λx.(x(λy.(y x)))) as λx.x(λy.y x). x # M means that the variable x does not occur anywhere in the λ-term M.

6 L9 106 Free and bound variables In λx.m, we call x the bound variable and M the body of the λ-abstraction. An occurrence of x in a λ-term M is called binding if in between λ and. (e.g. (λx.y x) x) bound if in the body of a binding occurrence of x (e.g. (λx.y x) x) free if neither binding nor bound (e.g. (λx.y x)x).

7 L9 106 Free and bound variables Sets of free and bound variables: FV(x) = {x} FV(λx.M) = FV(M) {x} FV(M N) = FV(M) FV(N) BV(x) = BV(λx.M) = BV(M) {x} BV(M N) = BV(M) BV(N) If FV(M) =, M is called a closed term, or combinator.

8 L9 106 Free and bound variables Sets of free and bound variables: FV(x) = {x} FV(λx.M) = FV(M) {x} FV(M N) = FV(M) FV(N) BV(x) = BV(λx.M) = BV(M) {x} BV(M N) = BV(M) BV(N) If FV(M) =, M is called a closed term, or combinator.

9 α-equivalence M = α M L9 107 λx.m is intended to represent the function f such that f(x) = M for all x. So the name of the bound variable is immaterial: if M = M{x /x} is the result of taking M and changing all occurrences of x to some variable x # M, then λx.m and λx.m both represent the same function. For example, λx.x and λy.y represent the same function (the identity function).

10 α-equivalence M = α M L9 107 is the binary relation inductively generated by the rules: x = α x z # (M N) M{z/x} = α N{z/y} λx.m = α λy.n M = α M N = α N M N = α M N where M{z/x} is M with all occurrences of x replaced by z.

11 α-equivalence M = α M L9 107 For example: λx.(λxx.x) x = α λy.(λx x.x)x because (λz x.z)x = α (λx x.x)x because λz x.z = α λx x.x and x = α x because λx.u = α λx.u and x = α x because u = α u and x = α x.

12 α-equivalence M = α M L9 107 For example: λx.(λxx.x) x = α λy.(λx x.x)x because (λz x.z)x = α (λx x.x)x because λz x.z = α λx x.x and x = α x because λx.u = α λx.u and x = α x because u = α u and x = α x.

13 α-equivalence M = α M L9 107 For example: λx.(λxx.x) x = α λy.(λx x.x)x because (λz x.z)x = α (λx x.x)x because λz x.z = α λx x.x and x = α x because λx.u = α λx.u and x = α x because u = α u and x = α x.

14 α-equivalence M = α M L9 107 For example: λx.(λxx.x) x = α λy.(λx x.x)x because (λz x.z)x = α (λx x.x)x because λz x.z = α λx x.x and x = α x because λx.u = α λx.u and x = α x because u = α u and x = α x.

15 α-equivalence M = α M L9 107 For example: λx.(λxx.x) x = α λy.(λx x.x)x because (λz x.z)x = α (λx x.x)x because λz x.z = α λx x.x and x = α x because λx.u = α λx.u and x = α x because u = α u and x = α x.

16 α-equivalence M = α M Fact: = α is an equivalence relation (reflexive, symmetric and transitive). We do not care about the particular names of bound variables, just about the distinctions between them. So α-equivalence classes of λ-terms are more important than λ-terms themselves. Textbooks (and these lectures) suppress any notation for α-equivalence classes and refer to an equivalence class via a representative λ-term (look for phrases like we identify terms up to α-equivalence or we work up to α-equivalence ). For implementations and computer-assisted reasoning, there are various devices for picking canonical representatives of α-equivalence classes (e.g. de Bruijn indexes, graphical representations,... ). L9 107

17 β-reduction Recall that λx.m is intended to represent the function f such that f (x) =M for all x. We can regard λx.m as a function on λ-terms via substitution: map each N to M[N/x]. So the natural notion of computation for λ-terms is given by stepping from a β-redex (λx.m)n to the corresponding β-reduct M[N/x] L10 109

18 Substitution N[M/x] x[m/x] = M y[m/x] = y if y = x (λy.n)[m/x] = λy.n[m/x] if y # (Mx) (N 1 N 2 )[M/x] = N 1 [M/x] N 2 [M/x] L9 108

19 Substitution N[M/x] x[m/x] = M y[m/x] = y if y = x (λy.n)[m/x] = λy.n[m/x] if y # (Mx) (N 1 N 2 )[M/x] = N 1 [M/x] N 2 [M/x] Side-condition y # (M x) (y does not occur in M and y = x) makessubstitution capture-avoiding. E.g. if x = y (λy.x)[y/x] = λy.y L9 108

20 Substitution N[M/x] x[m/x] = M y[m/x] = y if y = x (λy.n)[m/x] = λy.n[m/x] if y # (Mx) (N 1 N 2 )[M/x] = N 1 [M/x] N 2 [M/x] Side-condition y # (M x) (y does not occur in M and y = x) makessubstitution capture-avoiding. E.g. if x = y = z = x (λy.x)[y/x] = α (λz.x)[y/x] =λz.y In fact N N[M/x] induces a totally defined function from the set of α-equivalence classes of λ-terms to itself. L9 108

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27 β-reduction One-step β-reduction, M M : (λx.m)n M[N/x] M M MN M N M M λx.m λx.m M M NM NM N = α M M M M = α N N N L10 109

28 E.g. β-reduction ((λy.λz.z)u)y (λx.x y)((λy.λz.z)u) (λz.z)y y (λx.xy)(λz.z) L10 109

29 E.g. β-reduction ((λy.λz.z)u)y (λx.xy)((λy.λz.z)u) (λz.z)y y (λx.xy)(λz.z) L10 109

30 E.g. β-reduction ((λy.λz.z)u)y (λx.xy)((λy.λz.z)u) (λz.z)y y (λx.xy)(λz.z) L10 109

31 E.g. β-reduction ((λy.λz.z)u)y (λx.xy)((λy.λz.z)u) (λz.z)y y (λx.x y)(λz.z) L10 109

32 E.g. β-reduction ((λy.λz.z)u)y (λx.xy)((λy.λz.z)u) (λz.z)y y (λx.xy)(λz.z) L10 109

33 Many-step β-reduction, M M : M = α M M M (no steps) M M M M (1 step) M M M M M M (1 more step) E.g. (λx.xy)((λyz.z)u) y (λx.λy.x)y λz.y L10 110

34 β-conversion M = β N Informally: M = β N holds if N can be obtained from M by performing zero or more steps of α-equivalence, β-reduction, or β-expansion (= inverse of a reduction). E.g. u ((λxy. vx)y) = β (λx. ux)(λx. vy) because (λx. ux)(λx. vy) u(λx. vy) and so we have u ((λxy. vx)y) = α u ((λxy. vx)y) u(λy. vy) reduction = α u(λx. vy) (λx. ux)(λx. vy) expansion L10 111

35 β-conversion M = β N Informally: M = β N holds if N can be obtained from M by performing zero or more steps of α-equivalence, β-reduction, or β-expansion (= inverse of a reduction). E.g. u ((λxy. vx)y) = β (λx. ux)(λx. vy) because (λx. ux)(λx. vy) u(λx. vy) and so we have u ((λxy. vx)y) = α u ((λxy. vx)y) u(λy. vy) reduction = α u(λx. vy) (λx. ux)(λx. vy) expansion L10 111

36 β-conversion M = β N Informally: M = β N holds if N can be obtained from M by performing zero or more steps of α-equivalence, β-reduction, or β-expansion (= inverse of a reduction). E.g. u ((λxy. vx)y) = β (λx. ux)(λx. vy) because (λx. ux)(λx. vy) u(λx. vy) and so we have u ((λxy. vx)y) = α u ((λxy. vx)y) u(λy. vy) reduction = α u(λx. vy) (λx. ux)(λx. vy) expansion L10 111

37 β-conversion M = β N Informally: M = β N holds if N can be obtained from M by performing zero or more steps of α-equivalence, β-reduction, or β-expansion (= inverse of a reduction). E.g. u ((λxy. vx)y) = β (λx. ux)(λx. vy) because (λx. ux)(λx. vy) u(λx. vy) and so we have u ((λxy. vx)y) = α u ((λxy. vx)y) u(λy. vy) reduction = α u(λx. vy) (λx. ux)(λx. vy) expansion L10 111

38 β-conversion M = β N is the binary relation inductively generated by the rules: M = α M M M M = β M M = β M M = β M M = β M M = β M M = β M M = β M M = β M λx.m = β λx.m M = β M N = β N MN= β M N L10 111